Prof. Enrique Mateus NievesPhD in Mathematics Education.1HIGHER ORDER DIFFERENTIAL EQUATIONSHomogeneous linear equations with constant coefficients of order two andhigher.Apply reduction method to determine a solution of the nonhomogeneous equation given in thefollowing exercises. The indicated function y1(x), is a solution of the associated homogeneousequation. Determine a second solution of the homogeneous equation and a particular solutionof the inhomogeneous ED1.x1 ey;yy 224 2. 11 1y;yy3.x1xey;eyyy 3523 4.x1 ey;xyyy 34Problems for group discussion:1. Make a convincing demonstration that the second order equation ;cyybyu 0a, b, c, constant always has at least one solution of the formxm1 ey 1 , where 1m isa constant.2. Two. Explain why E. D. 1st point must have, consequently, a second solution of theformxm2 ey 2 or formxm2 xey 2 , where 1m y 2m are constants.HOMOGENEOUS LINEAR EQUATIONS WITH CONSTANT COEFFICIENTSWe have seen that the first order linear equation, 0 uydxdy, where a is a constant, has theexponential solutionax1 ecy 1 ranging ;.- therefore as natural to try to determine ifthere are exponential solutions .- homogeneous linear equations of higher order type: 001211 yayayayaya nnnn (1)Where the coefficients ,1,0,i,ai n are real constants and 0na . To our surprise, allsolutions of the equation (1) are exponential functions or are formed from exponentialfunctions.
Prof. Enrique Mateus NievesPhD in Mathematics Education.2Method of solution: start with the special case of the second order equation ay” + by’ + cy =0. (2) If we try a solution of the formmxey , thensmxmey andmxemy 2 , so thatthe equation (2) becomes:, 0 mxmxmx2cebmeeam , or 02 cbmamemx. Asmxe never zero when x has real value, the only way that theexponential function satisfies the differential equation is choosing a m such that it is a root ofthe quadratic equation 02 cbmam (3).This equation is called auxiliary equation or characteristic equation of the differential equation(2). Examine three cases: the solutions of the auxiliary equation corresponding to distinct realroots, real and equal roots and complex conjugate roots.CASE I: distinct real roots:If equation (3) has two distinct real roots, 1m y 2m , arrived at two solutions,xm1 ey 1 andxm2 ey 2 . These functions are linearly independent on .- and, therefore, form afundamental set. Then, the general solution of equation (2) in this interval isxmxmH ececy 2121 (4)CASE II: Real Estate and equalWhen 21 mm we necessarily exponential only solution,xm1 ey 1 . According to thequadratic formula,abm12 because the only way 21 mm is 04 acb2. Thus, a secondsolution of the equation is: xmxmxmxmxmH xedxeeeey 1111122(5)In this equation we take thatab2m1 . The general solution is thereforexmxmH xececy 1121 (6)
Prof. Enrique Mateus NievesPhD in Mathematics Education.3CASE III: complex conjugate roots.If 21 m,,m are complex, we can write im1 y im2 , where0and and p > 0 and they are real, and e .i21 There is no formal differencebetween this case and case I, hence,x)i(ececy x)i(H 21 However, inpractice it is preferred to work with real functions and not complex exponential. With this objectusing Eulers formula: ,senicosei that is a real number. The consequence ofthis formula is that: ,xsenixcose xi and ,xsenixcose x-i (7) where wehave used x)(cosx)(-cos and x)(senx)(-sen . Note that if the first add andthen subtract the two equations (7), we obtain respectively:,xcosee x-ixi2 y .xsiniee x-ixi2Asx)i(ececy x)i( 21 is a solution to equation (2) for any choice of theconstants 1c and, 2c if 121 cc and ,c 11 12 c obtain the solutions:x)i(eey x)i(1 yx)i(eey x)i(2 But xcoseeeey xxixix1 2 and xsinieeeey xxixix2 2 Accordingly, the results demonstrate that the last two real functions xcose xandxsene xare solutions of the equation (2). Moreover, these solutions form a fundamental,therefore .- , the general solution is: xsincxcoscexsinecxcosecy2xxxH121Second order differential equationsSolve the following differential equations:1. 0352 yyy
Prof. Enrique Mateus NievesPhD in Mathematics Education.4SOLUTION: I present the auxiliary equations, roots and corresponding general solutions. 33120352 2112 2m;mmmmm Hence:xececyx32122. 02510 yyySOLUTION: 50502510 122 2mmmmm Hence:xxxececy 5251 3. 0 yyySOLUTION: ,i21-m,i21-mmm 2232301 12 Hence:x23sincx23cosceyxH 2124. Initial value problem. Solve the initial value problem 20134 0y-1,(0)y;yyySOLUTION: The roots of the auxiliary equationi32m,i32mmm 2 120134 so that x3sincx3coscey xH 212By applying the condition 10 )(y , we see that 0senc0cosce 2101 and11 c . We differentiate the above equation and then applying y’(O) = 2 we get232 2 c or342 c ; therefore, the solution is: x3senx3cosey xH342
Prof. Enrique Mateus NievesPhD in Mathematics Education.5The two differential equations, 0 pyy y 02 kyy , k real, are important inapplied mathematics. For the first, the auxiliary equation 022 km has imaginary rootsikm1 y ikm1 . According to equation (8), with 0 y k, the general solution iskxsenckxcoscy 2 1 (9)Auxiliary equation the second equation, 022 km , has distinct real roots km1 ykm1 ; therefore its general solution is-kx2kxececy 1 (10)Note that if we choose 212 cc1 and then 21221 cyc1 in ,10 particular solutions wekxcosheey-kxkx2and kx.senheey-kxkx2inasmuch as kxcosh andkxsenh are linearly independent in any range of the x axis, an alternative form of thegeneral solution of 0 pyy is kxsenhckxcoshcy 2Higher-Order EquationsIn general, to solve a differential equation of order n as 001211 yayayayaya nnnn (11)Where n,20,1,i,ai are real constants, we must solve a polynomial equation of degreen: 0012211 amamamama nnnn (12)If all the roots of equation (12) are real and distinct, the general solution of equation (11) is.ecececy xmnxm2xm n21 1Its difficult to summarize the analogous cases II and III because the roots of an auxiliaryequation of degree greater than two can occur in many combinations. For example, a quinticcould have five distinct real roots, or three distinct real roots and two complex, or four real andcomplex, five reals but equal, but two equals five reals, and so on. When 1m is a root of anequation k multiplicity auxiliary degree n (ie roots equals k), one can show that the solutionsare linearly independent
Prof. Enrique Mateus NievesPhD in Mathematics Education.6.exex,xe,e xmkxmxmxm 1111 12 Finally, remember that when the coefficients are real, complex roots of auxiliary equationalways appear in conjugate pairs. Thus, for example, a cubic polynomial equation may havetwo complex zeros at most.Third-order differential equationResolve 043 yyy y”’ + 3~” - 4y = 0.SOLUTION: In reviewing 043 23 mm we should note that one of its roots is 11m .If we divide 43 23 mm eight ,m 1 we see that ,mmmmmmm 22144143 223 and then the other roots are2 32 mm . Thus, the general solution is.xecececy -2x-2x2x31 Fourth-order differential equationResolve 02 2244 ydxyddxydSOLUTION: The auxiliary equation is 010132224 mmm and has theroots imm 31 y imm 42 . Thus, according to the case II, the solution is:.ecxecececy -ix4ix-ix2ix 31According to Eulers formula, we can write the grouping-ix2ixecec 1 in the formxsencxcosc 21 With a change in the definition of the constants. equally, -ix4ixececx 3 can be expressed in the form xsencxcoscx 43 Accordingly, thegeneral solution is.x.sinxcxcosxcxsincxcoscy 42 31
Prof. Enrique Mateus NievesPhD in Mathematics Education.7General Exercises Nonhomogeneous linear equations with constant coefficientsof order two and higher.For each of the following E. D. finds the general solution:1. 04 yy 2. 052 yy3. 036 yy 4. 08 yy5. 09 yy 6. 03 yy7. 06 yyy 8. 023 yyy9. 016822 ydxdydxyd10. 0251022 ydxdydxyd11. 053 yyy 12. 048 yyy13. 02512 yyy 14. 028 yyy15. 054 yyy 16 0432 yyy17. 023 yyy 18. 022 yyy19. 054 yyy 20. 044 yyy21. 0 yy 22. 05 yy23. 0935 yyyy 24. 01243 yyyy25. 02 yyy 26. 04 yyy27. 09223344 ydxyddxyddxyd28. 051025 22334455 ydxdydxyddxyddxyddxyd